July  2012, 8(3): 691-703. doi: 10.3934/jimo.2012.8.691

Upper Hölder estimates of solutions to parametric primal and dual vector quasi-equilibria

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331

Received  March 2011 Revised  January 2012 Published  June 2012

In this paper, on one hand, we discuss upper Hölder type estimates of solutions to parametric vector quasi-equilibria with general settings, which generalize and extend the results of Chen et al. (Optim. Lett. 5: 85-98, 2011). On the other hand, combining the technique used for primal problems with suitable modifications, we also study upper Hölder type estimates of solutions to Minty-type parametric dual vector quasi-equilibria. The consequences obtained for dual problems are new in the literature.
Citation: Chunrong Chen, Shengji Li. Upper Hölder estimates of solutions to parametric primal and dual vector quasi-equilibria. Journal of Industrial & Management Optimization, 2012, 8 (3) : 691-703. doi: 10.3934/jimo.2012.8.691
References:
[1]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems,, J. Math. Anal. Appl., 294 (2004), 699. doi: 10.1016/j.jmaa.2004.03.014.

[2]

L. Q. Anh and P. Q. Khanh, On the stability of the solution sets of general multivalued vector quasiequilibrium problems,, J. Optim. Theory Appl., 135 (2007), 271. doi: 10.1007/s10957-007-9250-9.

[3]

L. Q. Anh and P. Q. Khanh, On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems,, J. Math. Anal. Appl., 321 (2006), 308. doi: 10.1016/j.jmaa.2005.08.018.

[4]

L. Q. Anh and P. Q. Khanh, Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces,, J. Glob. Optim., 37 (2007), 449. doi: 10.1007/s10898-006-9062-8.

[5]

L. Q. Anh and P. Q. Khanh, Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions,, J. Glob. Optim., 42 (2008), 515. doi: 10.1007/s10898-007-9268-4.

[6]

L. Q. Anh and P. Q. Khanh, Sensitivity anlysis for weak and strong vector quasiequilibrium problems,, Vietnam J. Math., 37 (2009), 237.

[7]

Q. H. Ansari, A. H. Siddiqi and S. Y. Wu, Existence and duality of generalized vector equilibrium problems,, J. Math. Anal. Appl., 259 (2001), 115. doi: 10.1006/jmaa.2000.7397.

[8]

E. M. Bednarczuk, Strong pseudomonotonicity, sharp efficiency and stability for parametric vector equilibria,, in, 17 (2007), 9. doi: 10.1051/proc:071702.

[9]

E. M. Bednarczuk, Weak sharp efficiency and growth condition for vector-valued functions with applications,, Optimization, 53 (2004), 455. doi: 10.1080/02331930412331330478.

[10]

M. Bianchi and R. Pini, Sensitivity for parametric vector equilibria,, Optimization, 55 (2006), 221. doi: 10.1080/02331930600662732.

[11]

J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems,", Springer Series in Operations Research, (2000).

[12]

G.-Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization: Set-Valued and Variational Analysis,", Lecture Notes in Economics and Mathematical Systems, 541 (2005).

[13]

C. R. Chen and S. J. Li, Semicontinuity of the solution set map to a set-valued weak vector variational inequality,, J. Ind. Manag. Optim., 3 (2007), 519.

[14]

C. R. Chen, S. J. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems,, J. Glob. Optim., 45 (2009), 309. doi: 10.1007/s10898-008-9376-9.

[15]

C. R. Chen, S. J. Li, J. Zeng and X. B. Li, Error analysis of approximate solutions to parametric vector quasiequilibrium problems,, Optim. Lett., 5 (2011), 85.

[16]

F. Giannessi, ed., "Vector Variational Inequalities and Vector Equilibria: Mathematical Theories,", Nonconvex Optimization and its Applications, 38 (2000).

[17]

X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems,, J. Optim. Theory Appl., 138 (2008), 197. doi: 10.1007/s10957-008-9378-2.

[18]

X. H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems,, J. Optim. Theory Appl., 139 (2008), 35. doi: 10.1007/s10957-008-9429-8.

[19]

N. J. Huang, J. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems,, Math. Comput. Modelling., 43 (2006), 1267. doi: 10.1016/j.mcm.2005.06.010.

[20]

K. Kimura and J.-C. Yao, Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems,, J. Glob. Optim., 41 (2008), 187. doi: 10.1007/s10898-007-9210-9.

[21]

K. Kimura and J. C. Yao, Semicontinuity of solution mappings of parametric generalized vector equilibrium problems,, J. Optim. Theory Appl., 138 (2008), 429. doi: 10.1007/s10957-008-9386-2.

[22]

K. Kimura and J.-C. Yao, Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems,, J. Ind. Manag. Optim., 4 (2008), 167.

[23]

I. V. Konnov and S. Schaible, Duality for equilibrium problems under generalized monotonicity,, J. Optim. Theory Appl., 104 (2000), 395. doi: 10.1023/A:1004665830923.

[24]

S. J. Li, G. Y. Chen and K. L. Teo, On the stability of generalized vector quasivariational inequality problems,, J. Optim. Theory Appl., 113 (2002), 283. doi: 10.1023/A:1014830925232.

[25]

S. J. Li and Z. M. Fang, On the stability of a dual weak vector variational inequality problem,, J. Ind. Manag. Optim., 4 (2008), 155.

[26]

S. J. Li and C. R. Chen, Stability of weak vector variational inequality,, Nonlinear Anal., 70 (2009), 1528. doi: 10.1016/j.na.2008.02.032.

[27]

S.-J. Li, H.-M. Liu and C.-R. Chen, Lower semicontinuity of parametric generalized weak vector equilibrium problems,, Bull. Aust. Math. Soc., 81 (2010), 85. doi: 10.1017/S0004972709000628.

[28]

S. J. Li, X .B. Li and K. L. Teo, The Hölder continuity of solutions to generalized vector equilibrium problems,, European J. Oper. Res., 199 (2009), 334. doi: 10.1016/j.ejor.2008.12.024.

[29]

S. J. Li, C. R. Chen, X. B. Li and K. L. Teo, Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems,, European J. Oper. Res., 210 (2011), 148. doi: 10.1016/j.ejor.2010.10.005.

[30]

P. H. Sach, L. A. Tuan and G. M. Lee, Sensitivity results for a general class of generalized vector quasi-equilibrium problems with set-valued maps,, Nonlinear Anal., 71 (2009), 571. doi: 10.1016/j.na.2008.10.098.

[31]

A. Shapiro, Perturbation analysis of optimization problems in Banach spaces,, Numer. Funct. Anal. Optim., 13 (1992), 97. doi: 10.1080/01630569208816463.

[32]

M. M. Wong, Lower semicontinuity of the solution map to a parametric vector variational inequality,, J. Glob. Optim., 46 (2010), 435. doi: 10.1007/s10898-009-9447-6.

show all references

References:
[1]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems,, J. Math. Anal. Appl., 294 (2004), 699. doi: 10.1016/j.jmaa.2004.03.014.

[2]

L. Q. Anh and P. Q. Khanh, On the stability of the solution sets of general multivalued vector quasiequilibrium problems,, J. Optim. Theory Appl., 135 (2007), 271. doi: 10.1007/s10957-007-9250-9.

[3]

L. Q. Anh and P. Q. Khanh, On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems,, J. Math. Anal. Appl., 321 (2006), 308. doi: 10.1016/j.jmaa.2005.08.018.

[4]

L. Q. Anh and P. Q. Khanh, Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces,, J. Glob. Optim., 37 (2007), 449. doi: 10.1007/s10898-006-9062-8.

[5]

L. Q. Anh and P. Q. Khanh, Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions,, J. Glob. Optim., 42 (2008), 515. doi: 10.1007/s10898-007-9268-4.

[6]

L. Q. Anh and P. Q. Khanh, Sensitivity anlysis for weak and strong vector quasiequilibrium problems,, Vietnam J. Math., 37 (2009), 237.

[7]

Q. H. Ansari, A. H. Siddiqi and S. Y. Wu, Existence and duality of generalized vector equilibrium problems,, J. Math. Anal. Appl., 259 (2001), 115. doi: 10.1006/jmaa.2000.7397.

[8]

E. M. Bednarczuk, Strong pseudomonotonicity, sharp efficiency and stability for parametric vector equilibria,, in, 17 (2007), 9. doi: 10.1051/proc:071702.

[9]

E. M. Bednarczuk, Weak sharp efficiency and growth condition for vector-valued functions with applications,, Optimization, 53 (2004), 455. doi: 10.1080/02331930412331330478.

[10]

M. Bianchi and R. Pini, Sensitivity for parametric vector equilibria,, Optimization, 55 (2006), 221. doi: 10.1080/02331930600662732.

[11]

J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems,", Springer Series in Operations Research, (2000).

[12]

G.-Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization: Set-Valued and Variational Analysis,", Lecture Notes in Economics and Mathematical Systems, 541 (2005).

[13]

C. R. Chen and S. J. Li, Semicontinuity of the solution set map to a set-valued weak vector variational inequality,, J. Ind. Manag. Optim., 3 (2007), 519.

[14]

C. R. Chen, S. J. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems,, J. Glob. Optim., 45 (2009), 309. doi: 10.1007/s10898-008-9376-9.

[15]

C. R. Chen, S. J. Li, J. Zeng and X. B. Li, Error analysis of approximate solutions to parametric vector quasiequilibrium problems,, Optim. Lett., 5 (2011), 85.

[16]

F. Giannessi, ed., "Vector Variational Inequalities and Vector Equilibria: Mathematical Theories,", Nonconvex Optimization and its Applications, 38 (2000).

[17]

X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems,, J. Optim. Theory Appl., 138 (2008), 197. doi: 10.1007/s10957-008-9378-2.

[18]

X. H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems,, J. Optim. Theory Appl., 139 (2008), 35. doi: 10.1007/s10957-008-9429-8.

[19]

N. J. Huang, J. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems,, Math. Comput. Modelling., 43 (2006), 1267. doi: 10.1016/j.mcm.2005.06.010.

[20]

K. Kimura and J.-C. Yao, Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems,, J. Glob. Optim., 41 (2008), 187. doi: 10.1007/s10898-007-9210-9.

[21]

K. Kimura and J. C. Yao, Semicontinuity of solution mappings of parametric generalized vector equilibrium problems,, J. Optim. Theory Appl., 138 (2008), 429. doi: 10.1007/s10957-008-9386-2.

[22]

K. Kimura and J.-C. Yao, Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems,, J. Ind. Manag. Optim., 4 (2008), 167.

[23]

I. V. Konnov and S. Schaible, Duality for equilibrium problems under generalized monotonicity,, J. Optim. Theory Appl., 104 (2000), 395. doi: 10.1023/A:1004665830923.

[24]

S. J. Li, G. Y. Chen and K. L. Teo, On the stability of generalized vector quasivariational inequality problems,, J. Optim. Theory Appl., 113 (2002), 283. doi: 10.1023/A:1014830925232.

[25]

S. J. Li and Z. M. Fang, On the stability of a dual weak vector variational inequality problem,, J. Ind. Manag. Optim., 4 (2008), 155.

[26]

S. J. Li and C. R. Chen, Stability of weak vector variational inequality,, Nonlinear Anal., 70 (2009), 1528. doi: 10.1016/j.na.2008.02.032.

[27]

S.-J. Li, H.-M. Liu and C.-R. Chen, Lower semicontinuity of parametric generalized weak vector equilibrium problems,, Bull. Aust. Math. Soc., 81 (2010), 85. doi: 10.1017/S0004972709000628.

[28]

S. J. Li, X .B. Li and K. L. Teo, The Hölder continuity of solutions to generalized vector equilibrium problems,, European J. Oper. Res., 199 (2009), 334. doi: 10.1016/j.ejor.2008.12.024.

[29]

S. J. Li, C. R. Chen, X. B. Li and K. L. Teo, Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems,, European J. Oper. Res., 210 (2011), 148. doi: 10.1016/j.ejor.2010.10.005.

[30]

P. H. Sach, L. A. Tuan and G. M. Lee, Sensitivity results for a general class of generalized vector quasi-equilibrium problems with set-valued maps,, Nonlinear Anal., 71 (2009), 571. doi: 10.1016/j.na.2008.10.098.

[31]

A. Shapiro, Perturbation analysis of optimization problems in Banach spaces,, Numer. Funct. Anal. Optim., 13 (1992), 97. doi: 10.1080/01630569208816463.

[32]

M. M. Wong, Lower semicontinuity of the solution map to a parametric vector variational inequality,, J. Glob. Optim., 46 (2010), 435. doi: 10.1007/s10898-009-9447-6.

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